2.7.2: Many-Sorted Logic

\(\def\Assign#1#2{ { #1^{\Struct{#2}} } }\)
\(\def\Atom#1#2{ { \mathord{#1}(#2) } }\)
\(\def\Bin{ {\mathbb{B}} }\)
\(\def\cardeq#1#2{ { #1 \approx #2 } }\)
\(\def\cardle#1#2{ { #1 \preceq #2 } }\)
\(\def\cardless#1#2{ { #1 \prec #2 } }\)
\(\def\cardneq#1#2{ { #1 \not\approx #2 } }\)
\(\def\comp#1#2{ { #2 \circ #1 } }\)
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\(\def\eq[#1][#2]{ { #1 = #2 } }\)
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\(\def\equivclass#1#2{ { #1/_{#2} } }\)
\(\def\equivrep#1#2{ { [#1]_{#2} } }\)
\(\def\Exchange{ { \text{X} } }\)
\(\def\False{ { \mathbb{F} } }\)
\(\def\FalseCl{ { \lfalse_C } }\)
\(\def\FalseInt{ { \lfalse_I } }\)
\(\def\fCenter{ { \,\Sequent\, } }\)
\(\def\fdefined{ { \;\downarrow } }\)
\(\def\fn#1{ { \operatorname{#1} } }\)
\(\def\Frm[#1]{ {\operatorname{Frm}(\Lang #1)} }\)
\(\def\fundefined{ { \;\uparrow } }\)
\(\def\funimage#1#2{ { #1[#2] } }\)
\(\def\funrestrictionto#1#2{ { #1 \restriction_{#2} } }\)
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\(\def\indfrm{ { A } }\)
\(\def\indfrmp{ { A } }\)
\(\def\joinrel{\mathrel{\mkern-3mu}}\)
\(\def\lambd[#1][#2]{\lambda #1 . #2}\)
\(\def\Lang#1{ { \mathcal{#1} } }\)
\(\def\LeftR#1{ { {#1}\mathrm{L} } }\)
\(\def\len#1{ {\operatorname{len}(#1)} }\)
\(\def\lexists#1#2{ { \exists #1\, #2 } }\)
\(\def\lfalse{ {\bot} }\)
\(\def\lforall#1#2{ { \forall#1\, #2 } }\)
\(\newcommand{\lif}{\rightarrow}\)
\(\newcommand{\liff}{\leftrightarrow}\)
\(\def\Log#1{ { \mathbf{#1} } }\)
\(\def\ltrue{ {\top} }\)
\(\def\Id#1{ {\operatorname{Id}_#1} }\)
\(\def\Int{ {\mathbb{Z}} }\)
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\(\def\Nat{ {\mathbb{N}} }\)
\(\def\nicefrac#1#2{ {{}^#1/_#2} }\)
\(\def\num#1{ { \overline{#1} } }\)

\(\def\ran#1{ {\operatorname{ran}(#1)} }\)
\(\newcommand{\Obj}[1]{\mathsf{#1}}\)
\(\def\Rat{ {\mathbb{Q}} }\)
\(\def\Real{ {\mathbb{R}} }\)
\(\def\RightR#1{ { {#1}\mathrm{R} } }\)
\(\def\Part#1#2{ { \Atom{\Obj P}{#1, #2} } }\)
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\(\def\Relbar{\mathrel{=}}\)
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\(\def\TMtrans#1#2#3{ { #1,#2,#3 } }\)
\(\def\Trm[#1]{ {\operatorname{Trm}(\Lang #1)} }\)
\(\def\True{ { \mathbb{T} } }\)
\(\newcommand{\TRule}[2]{#2#1}\)
\(\def\tuple#1{ {\langle #1 \rangle} }\)
\(\newcommand{\Value}[3][\,]{\mathrm{Val}_{#1}^{#3}(#2)}\)
\(\def\Var{ { \mathrm{Var} } }\)
\(\newcommand{\varAssign}[3]{#1 \sim_{#3} #2}\)
\(\def\Weakening{ { \text{W} } }\)

In first-order logic, variables and quantifiers range over a single domain. But it is often useful to have multiple (disjoint) domains: for example, you might want to have a domain of numbers, a domain of geometric objects, a domain of functions from numbers to numbers, a domain of abelian groups, and so on.

Many-sorted logic provides this kind of framework. One starts with a list of “sorts”—the “sort” of an object indicates the “domain” it is supposed to inhabit. One then has variables and quantifiers for each sort, and (usually) an identity predicate for each sort. Functions and relations are also “typed” by the sorts of objects they can take as arguments. Otherwise, one keeps the usual rules of first-order logic, with versions of the quantifier-rules repeated for each sort.

For example, to study international relations we might choose a language with two sorts of objects, French citizens and German citizens. We might have a unary relation, “drinks wine,” for objects of the first sort; another unary relation, “eats wurst,” for objects of the second sort; and a binary relation, “forms a multinational married couple,” which takes two arguments, where the first argument is of the first sort and the second argument is of the second sort. If we use variables \(a\), \(b\), \(c\) to range over French citizens and \(x\), \(y\), \(z\) to range over German citizens, then \[\lforall{a}{\lforall{}{} x}[(\Atom{\Obj{MarriedTo}}{a,x} \lif (\Atom{\Obj{DrinksWine}}{a} \lor \lnot \Atom{\Obj{EatsWurst}}{x})]]\nonumber\] asserts that if any French person is married to a German, either the French person drinks wine or the German doesn’t eat wurst.

Many-sorted logic can be embedded in first-order logic in a natural way, by lumping all the objects of the many-sorted domains together into one first-order domain, using unary predicate symbols to keep track of the sorts, and relativizing quantifiers. For example, the first-order language corresponding to the example above would have unary predicate symbols “\(\Obj{German}\)” and “\(\Obj{French}\),” in addition to the other relations described, with the sort requirements erased. A sorted quantifier \(\lforall{x}{A}\), where \(x\) is a variable of the German sort, translates to \[\lforall{x}{(\Atom{\Obj{German}}{x} \lif A)}.\nonumber\] We need to add axioms that insure that the sorts are separate—e.g., \(\lforall{x}{\lnot (\Atom{\Obj{German}}{x} \land \Atom{\Obj{French}}{x})}\)—as well as axioms that guarantee that “drinks wine” only holds of objects satisfying the predicate \(\Atom{\Obj{French}}{x}\), etc. With these conventions and axioms, it is not difficult to show that many-sorted sentences translate to first-order sentences, and many-sorted derivations translate to first-order derivations. Also, many-sorted structures “translate” to corresponding first-order structures and vice-versa, so we also have a completeness theorem for many-sorted logic.